We may equip \(\mathcal {D}_X\)  with the middle perversity \(t\) -structure \(({\mathfrak {p}}{}\mathcal {D}_X^{\le 0}, {\mathfrak {p}}{}\mathcal {D}_X^{\ge 0})\) . The truncations with respect to this \(t\) -structure are denoted \({\mathfrak {p}}{}\tau _{\ge k}\colon \mathcal {D}_X \rightarrow {\mathfrak {p}}{}\mathcal {D}_X^{\ge k}\)  and \({\mathfrak {p}}{}\tau _{\le k}\colon \mathcal {D}_X \rightarrow {\mathfrak {p}}{}\mathcal {D}_X^{\le k}\) , and its heart \(\mathcal {P}_X := {\mathfrak {p}}{}\mathcal {D}_X^{\le 0} \cap {\mathfrak {p}}{}\mathcal {D}_X^{\ge 0}\)  is the abelian category of perverse sheaves on \(X\) . With this notation we have the perverse cohomology functors \({\mathfrak {p}}{}{\mathcal {H}}^k \colon \mathcal {D}_X \rightarrow \mathcal {P}_X\)  defined by \({\mathfrak {p}}{}{\mathcal {H}}^k := {\mathfrak {p}}{}\tau _{\le 0} \circ {\mathfrak {p}}{}\tau _{\ge 0} \circ [k]\) , where \([k]\)  denotes the usual shift functor. For more details of these constructions, we refer the reader to [1]}.
